If the polynomials (2x^3 + ax^2 + 3x − 5) and (x^3 + x^2 − 2x + a) leave the same remainder when divided by (x − 2), find the value of a. Also, find the remainder in each case.

Question

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By remainder theorem, If f(x) is divided by (x - a), then remainder = f(a). Let, p(x) = 2x3 + ax2 + 3x - 5 and q(x) = x3 + x2 - 2x + a Divisor : ⇒ x - 2 ⇒ x = 2 ⇒ p(2) = 2(2)3 + a(2)2 + 3(2) - 5 = 2(8) + 4a + 6 - 5 = 16 + 4a + 1 = 4a + 17. ⇒ q(2) = (2)3 + (2)2 - 2(2) + a = 8 + 4 - 4 + a = 8 + a. Given, Polynomials 2x3 + ax2 + 3x - 5 and x3 + x2 - 2x + a leave the same remainder when divided by (x - 2). p(2) = q(2) ⇒ 4a + 17 = 8 + a ⇒ 4a - a = 8 - 17 ⇒ 3a = -9 ⇒ a = ⇒ a = -3. Substituting value of a in p(2) : ⇒ p(2) = 4a + 17 = 4(-3) + 17 = -12 + 17 = 5. Substituting value of a in q(2) : ⇒ q(2) = 8 + a = 8 - 3 = 5. Hence, the value of a = -3 and remainder in each case is 5.

Mathematics

If the polynomials 2x³ + ax² + 3x - 5 and x³ + x² - 2x + a leave the same remainder when divided by (x - 2), find the value of a. Also, find the remainder in each case.

Factorisation

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Mathematics

If the polynomials 2x3 + ax2 + 3x - 5 and x3 + x2 - 2x + a leave the same remainder when divided by (x - 2), find the value of a. Also, find the remainder in each case.

Factorisation

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